Question

What is the perimeter of △EFG?
E =(-5,7)
F=(7,0)
G=(-5,-5)
Perimeter = ______ Units

Answer 1

To find the perimeter of triangle \( \triangle EFG \) with vertices \( E(-5, 7) \), \( F(7, 0) \), and \( G(-5, -5) \), we first need to calculate the lengths of each side of the triangle.

1. **Calculate the length of side \( EF \)**:

\[
EF = \sqrt{(x_F - x_E)^2 + (y_F - y_E)^2}
\]
\[
EF = \sqrt{(7 - (-5))^2 + (0 - 7)^2} = \sqrt{(7 + 5)^2 + (0 - 7)^2}
\]
\[
EF = \sqrt{(12)^2 + (-7)^2} = \sqrt{144 + 49} = \sqrt{193}
\]

2. **Calculate the length of side \( FG \)**:

\[
FG = \sqrt{(x_G - x_F)^2 + (y_G - y_F)^2}
\]
\[
FG = \sqrt{(-5 - 7)^2 + (-5 - 0)^2} = \sqrt{(-12)^2 + (-5)^2}
\]
\[
FG = \sqrt{144 + 25} = \sqrt{169} = 13
\]

3. **Calculate the length of side \( EG \)**:

\[
EG = \sqrt{(x_G - x_E)^2 + (y_G - y_E)^2}
\]
\[
EG = \sqrt{(-5 - (-5))^2 + (-5 - 7)^2} = \sqrt{(0)^2 + (-12)^2} = \sqrt{0 + 144} = 12
\]

4. **Calculate the perimeter**:

The perimeter \( P \) of triangle \( EFG \) is the sum of the lengths of its sides:

\[
P = EF + FG + EG
\]
\[
P = \sqrt{193} + 13 + 12
\]
\[
P = \sqrt{193} + 25
\]

Thus, the perimeter of triangle \( EFG \) is approximately:

\[
P \approx 13.89 + 25 \approx 38.89 \text{ units}
\]

For the exact perimeter, we can leave it as:

\[
P = \sqrt{193} + 25 \text{ units}
\]